# A Geometric Perspective on Variational Autoencoders

arXiv: https://arxiv.org/abs/2209.07370 · HTML: https://mlsanity.com/paper/2209.07370/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 73 references that are in the map.
- First posted on arXiv 15 Sep 2022.

## Builds on (most cited first, 30 of 36)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Generating Diverse High-Fidelity Images with VQ-VAE-2](https://mlsanity.com/paper/1906.00446/) (2019)
- [Deep Variational Information Bottleneck](https://mlsanity.com/paper/1612.00410/) (2016)
- [InfoGAN: Interpretable Representation Learning by Information Maximizing Generative Adversarial Nets](https://mlsanity.com/paper/1606.03657/) (2016)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [Importance Weighted Autoencoders](https://mlsanity.com/paper/1509.00519/) (2015)
- [Autoencoding beyond pixels using a learned similarity metric](https://mlsanity.com/paper/1512.09300/) (2015)
- [Assessing Generative Models via Precision and Recall](https://mlsanity.com/paper/1806.00035/) (2018)
- [Wasserstein Auto-Encoders](https://mlsanity.com/paper/1711.01558/) (2017)
- [Adversarial Autoencoders](https://mlsanity.com/paper/1511.05644/) (2015)
- [VAE with a VampPrior](https://mlsanity.com/paper/1705.07120/) (2017)
- [Advances in Variational Inference](https://mlsanity.com/paper/1711.05597/) (2017)
- [Deep Unsupervised Clustering with Gaussian Mixture Variational Autoencoders](https://mlsanity.com/paper/1611.02648/) (2016)
- [Diagnosing and Enhancing VAE Models](https://mlsanity.com/paper/1903.05789/) (2019)
- [Hyperspherical Variational Auto-Encoders](https://mlsanity.com/paper/1804.00891/) (2018)
- [Variational Lossy Autoencoder](https://mlsanity.com/paper/1611.02731/) (2016)
- [Latent Space Oddity: on the Curvature of Deep Generative Models](https://mlsanity.com/paper/1710.11379/) (2017)
- [From Variational to Deterministic Autoencoders](https://mlsanity.com/paper/1903.12436/) (2019)
- [Inference Suboptimality in Variational Autoencoders](https://mlsanity.com/paper/1801.03558/) (2018)
- [Continuous Hierarchical Representations with Poincar'e Variational Auto-Encoders](https://mlsanity.com/paper/1901.06033/) (2019)
- [Learning Latent Space Energy-Based Prior Model](https://mlsanity.com/paper/2006.08205/) (2020)
- [VAEBM: A Symbiosis between Variational Autoencoders and Energy-based Models](https://mlsanity.com/paper/2010.00654/) (2020)
- [Variational Autoencoders Pursue PCA Directions (by Accident)](https://mlsanity.com/paper/1812.06775/) (2018)
- [The Riemannian Geometry of Deep Generative Models](https://mlsanity.com/paper/1711.08014/) (2017)
- [Explorations in Homeomorphic Variational Auto-Encoding](https://mlsanity.com/paper/1807.04689/) (2018)
- [Resampled Priors for Variational Autoencoders](https://mlsanity.com/paper/1810.11428/) (2018)
- [Metrics for Deep Generative Models](https://mlsanity.com/paper/1711.01204/) (2017)
- [Learning Hierarchical Priors in VAEs](https://mlsanity.com/paper/1905.04982/) (2019)

## Built on by (most cited first, 24 of 24)

- [Metric Flow Matching for Smooth Interpolations on the Data Manifold](https://mlsanity.com/paper/2405.14780/) (2024)
- [Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connections](https://mlsanity.com/paper/2404.02954/) (2024)
- [VTAE: Variational Transformer Autoencoder with Manifolds Learning](https://mlsanity.com/paper/2304.00948/) (2023)
- [The Gaussian Discriminant Variational Autoencoder (GdVAE): A Self-Explainable Model with Counterfactual Explanations](https://mlsanity.com/paper/2409.12952/) (2024)
- [Connecting Neural Models Latent Geometries with Relative Geodesic Representations](https://mlsanity.com/paper/2506.01599/) (2025)
- [Ricci flow regularization in latent spaces for the forward learning of partial differential equations](https://mlsanity.com/paper/2401.14591/) (2024)
- [Latent Space Characterization of Autoencoder Variants](https://mlsanity.com/paper/2412.04755/) (2024)
- [Latent Diffusion Inversion Requires Understanding the Latent Space](https://mlsanity.com/paper/2511.20592/) (2025)
- [A mathematical framework of intelligence and consciousness based on Riemannian Geometry](https://mlsanity.com/paper/2407.11024/) (2024)
- [Local EGOP for Continuous Index Learning](https://mlsanity.com/paper/2601.07061/) (2026)
- [Variational Inference for Longitudinal Data Using Normalizing Flows](https://mlsanity.com/paper/2303.14220/) (2023)
- [Learning Pose Image Manifolds Using Geometry-Preserving GANs and Elasticae](https://mlsanity.com/paper/2305.10513/) (2023)
- [Illumination and Shadows in Head Rotation: experiments with Denoising Diffusion Models](https://mlsanity.com/paper/2308.06057/) (2023)
- [Sampling From Autoencoders' Latent Space via Quantization And Probability Mass Function Concepts](https://mlsanity.com/paper/2308.10704/) (2023)
- [Variational autoencoders with latent high-dimensional steady geometric flows for dynamics](https://mlsanity.com/paper/2410.10137/) (2024)
- [Disentanglement as Identifiable Pushforward Factorisation](https://mlsanity.com/paper/2410.22559/) (2024)
- [Assessing local deformation and computing scalar curvature with nonlinear conformal regularization of decoders](https://mlsanity.com/paper/2508.20413/) (2025)
- [Privacy-Utility Trade-off in Data Publication: A Bilevel Optimization Framework with Curvature-Guided Perturbation](https://mlsanity.com/paper/2509.02048/) (2025)
- [A Geometric Unification of Generative AI with Manifold-Probabilistic Projection Models](https://mlsanity.com/paper/2510.00666/) (2025)
- [Complex variational autoencoders admit K"ahler structure](https://mlsanity.com/paper/2511.15172/) (2025)
- [Hyperparameter Trajectory Inference with Conditional Lagrangian Optimal Transport](https://mlsanity.com/paper/2603.01771/) (2026)
- [Manifold-Matching Autoencoders](https://mlsanity.com/paper/2603.16568/) (2026)
- [Geometric Decoupling: Diagnosing the Structural Instability of Latent](https://mlsanity.com/paper/2604.18804/) (2026)
- [An Elastic Shape Variational Autoencoder for Skeleton Pose Trajectories](https://mlsanity.com/paper/2605.09231/) (2026)

## Abstract (arXiv)

This paper introduces a new interpretation of the Variational Autoencoder framework by taking a fully geometric point of view. We argue that vanilla VAE models unveil naturally a Riemannian structure in their latent space and that taking into consideration those geometrical aspects can lead to better interpolations and an improved generation procedure. This new proposed sampling method consists in sampling from the uniform distribution deriving intrinsically from the learned Riemannian latent space and we show that using this scheme can make a vanilla VAE competitive and even better than more advanced versions on several benchmark datasets. Since generative models are known to be sensitive to the number of training samples we also stress the method's robustness in the low data regime.